Indices

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a^m means 'a' multiplied by itself m times. Here, 'a' is called the base and 'm' is called the power, exponent, or index. For example: 2^5 = 2 × 2 × 2 × 2 × 2 = 32

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a^m × a^n = a^(m+n). When multiplying powers with the same base, add the exponents. Example: 3^4 × 3^2 = 3^(4+2) = 3^6

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a^m ÷ a^n = a^(m-n). When dividing powers with the same base, subtract the exponents. Example: 5^7 ÷ 5^3 = 5^(7-3) = 5^4

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(a^m)^n = a^(mn). When raising a power to another power, multiply the exponents. Example: (2^3)^4 = 2^(3×4) = 2^12

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a^0 = 1 for any non-zero number a. Examples: 5^0 = 1, (-3)^0 = 1, (1/2)^0 = 1. This is one of the fundamental properties of exponents.

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a^(-n) = 1/a^n. A negative exponent means the reciprocal of the positive exponent. Example: 2^(-3) = 1/2^3 = 1/8. Also, 1/a^(-n) = a^n

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∜a = a^(1/n). The nth root of a number can be written as a fractional exponent. Examples: ∛8 = 8^(1/3) = 2, √16 = 16^(1/2) = 4

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a^(m/n) = ∜(a^m) = (∜a)^m. A fractional exponent can be written as a radical. Example: 8^(2/3) = ∛(8^2) = ∛64 = 4, or (∛8)^2 = 2^2 = 4